arbelos

The figure bounded by three semicircles AB, BC,
and AC, where ABC is a straight line. Archimedes
(about 250 BC) called it an "arbelos" – the Greek word for a knife
of the same shape used by shoemakers to cut and trim leather – and
wrote about it in his Liber assumptorum (Book of Lemmas).

Among the properties of the arbelos are that the sum of the two smaller
arc lengths is equal to the larger; the area of the arbelos is π/4 times
the product of the two smaller diameters (AB and BC); and
the area of the arbelos is equal to the area of a circle
whose diameter is the length of a perpendicular segment drawn from the tangent
point B of the two smaller semicircles to the point D where
it meets the larger semicircle.

The circles inscribed on each half of BD of the arbelos (called Archimedes's
circles) each have a diameter of (AB)(BC)/(AC).
Furthermore, the smallest circumcircle of these two circles has an area
equal to that of the arbelos.

Pappus of Alexandria wrote on the relations
of the chain of circles, C1, C2, C3, ..., (called a
Pappus chain or an arbelos train) that
are mutually tangent to the two larges semicircles and to each other. The
centers of these circles lie on an ellipse and the diameter of the nth
circle is (1/n) times the base of the perpendicular distance to the
base of the semicircle.